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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2016
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   Author: Urs
   Format: MarkdownItexI have started editing at _[[Thom's theorem]]_. So far it has just the definition of the bordism ring, the statement of the theorem and some literature.

   I have started editing at Thom’s theorem. So far it has just the definition of the bordism ring, the statement of the theorem and some literature.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2016
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   Author: Urs
   Format: MarkdownItexI have added a bunch of infrastructure (section _[Ingredients](https://ncatlab.org/nlab/show/Thom's+theorem#Ingredients)_) and then spelled out in detail the proof ([here](https://ncatlab.org/nlab/show/Thom's+theorem#PontrjaginThomConstructionGivesWellDefinedStableHomotopyGroup)) that the Pontrjagin-Thom construction first of all yields a well-defined function of sets $$ \left\{ {n\text{-}manifolds\;with\;stable} \atop {normal\;\mathcal{B}\text{-}structure} \right\} \longrightarrow \pi_n(M\mathcal{B}) \,. $$

   I have added a bunch of infrastructure (section Ingredients) and then spelled out in detail the proof (here) that the Pontrjagin-Thom construction first of all yields a well-defined function of sets

   $$\left\{ {n\text{-}manifolds\;with\;stable} \atop {normal\;\mathcal{B}\text{-}structure} \right\} \longrightarrow \pi_n(M\mathcal{B}) \,.$$
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2016
   • (edited May 31st 2016)
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   Author: Urs
   Format: MarkdownItexI have now spelled out the proof ([here](https://ncatlab.org/nlab/show/Thom's+theorem#PontrjaginThomIsRingHomomorphims)) that the PT construction yields a ring homomorphism $\Omega^\mathcal{B}_\bullet \to \pi_{\bullet}(M \mathcal{B})$, and the idea of the proof ([here](https://ncatlab.org/nlab/show/Thom's+theorem#MorphismFromCobordismRingToStableHomotopyGroupsOfThomSpectrumIsIso)) that this is an isomorphism.

   I have now spelled out the proof (here) that the PT construction yields a ring homomorphism $\Omega^\mathcal{B}_\bullet \to \pi_{\bullet}(M \mathcal{B})$, and the idea of the proof (here) that this is an isomorphism.